What do quadratic approximations look like | Summary and Q&A

TL;DR

Quadratic approximations provide a closer and more accurate estimation of a function by creating a surface that closely hugs the graph.

Key Insights

• 📈 Quadratic approximations create surfaces that closely follow the graph of a function.
• ❓ The surfaces resemble parabolas when sliced in any direction.
• 👋 Quadratic approximations require adjusting the coefficients of various terms to achieve the best fit to the graph.

Transcript

• [Voiceover] In the last couple videos I talked about the local linearization of a function. And in terms of graphs, there's a nice interpretation here where if you imagine a graph of a function and you want to approximate it near a specific point, you picture that point somewhere on the graph, and it doesn't have to be there, you know I can choos... Read More

Questions & Answers

Q: How does a quadratic approximation differ from local linearization?

While local linearization uses flat planes to approximate a function, quadratic approximations create surfaces that hug the graph more closely, resulting in a more accurate estimation.

Q: What does it mean for a surface to "hug" the graph?

When a surface closely hugs the graph, it means that it follows the curvature and shape of the graph closely, resulting in a more accurate representation of the function.

Q: How does the way a quadratic approximation hugs the graph change as we move around the point being approximated?

As we move around the point, the way a quadratic approximation hugs the graph can look different. It depends on the curvature and shape of the graph at that specific point, resulting in variations in the surface's appearance.

Q: How does a quadratic approximation provide a closer approximation than a linear function?

A quadratic approximation includes additional terms such as x squared, y squared, and x times y, allowing for more control and a closer fit to the graph compared to a linear function that only includes linear terms.

Summary & Key Takeaways

• Quadratic approximations take the concept of local linearization to the next level by creating a surface that closely follows the graph of a function.

• Unlike flat planes used in local linearization, quadratic approximations create surfaces that hug the graph more closely, resulting in a more accurate approximation.

• By slicing the surface in any direction, the resulting cross-section will resemble a parabola, allowing for a simpler representation of the function.